Math::...dRings(3pm)

📦
version 129 · 3 versions
, in debian bookworm, debian trixie, ubuntu jammy, ubuntu noble, ubuntu resolute, ubuntu stonking

💖 NAME 💖

Math::PlanePath::AztecDiamondRings — rings around an Aztec diamond shape 🦋

(and yes bestie, the header up top is Math::...dRings(3pm) from the User Contributed Perl Documentation, so you're literally in the right place 💅)

✨ SYNOPSIS ✨

use Math::PlanePath::AztecDiamondRings;
my $path = Math::PlanePath::AztecDiamondRings->new;
my ($x, $y) = $path->n_to_xy (123);

🌸 DESCRIPTION 🌸

Okay so this path makes rings around an Aztec diamond shape, and honestly the diagram is the whole vibe:

                 46-45                       4
                /     \
              47 29-28 44                    3
             /  /     \  \
           48 30 16-15 27 43  ...            2
          /  /  /     \  \  \  \
        49 31 17  7--6 14 26 42 62           1
       /  /  /  /     \  \  \  \  \
     50 32 18  8  2--1  5 13 25 41 61    <- Y=0
      |  |  |  |  |  |  |  |  |  |
     51 33 19  9  3--4 12 24 40 60          -1
       \  \  \  \     /  /  /  /
        52 34 20 10-11 23 39 59             -2
          \  \  \     /  /  /
           53 35 21-22 38 58                -3
             \  \     /  /
              54 36-37 57                   -4
                \     /
                 55-56                      -5

                     ^
    -5 -4 -3 -2 -1  X=0 1  2  3  4  5

So this one is similar to the DiamondSpiral, except all four corners get flattened to 2 vertical or horizontal, instead of just one corner like in the DiamondSpiral. It's only a small change to how the numbers line up along the sides, but it's more symmetric and honestly? we love that for it. ✨

The Y axis at N=1,6,15,28,45,66,etc are the hexagonal numbers k*(2k-1). The hexagonal numbers of the "second kind" 3,10,21,36,55,78, etc k*(2k+1), are the vertical at X=-1 going downwards. Combine those two and you get the triangular numbers 3,6,10,15,21,etc, k*(k+1)/2, alternately on one line and the other. Those are the positions of all the horizontal steps, ie. where dY=0. 🎀

The X axis at N=1,5,13,25,etc is the "centred square numbers". Those numbers come from drawing concentric squares with an extra point on each side each time. This path grows the exact same way, adding one extra point to each of the four sides. 🫶

*---*---*---*
|           |
| *---*---* |     count total "*"s for
| |       | |     centred square numbers
* | *---* | *
| | |   | | |
| * | * | * |
| | |   | | |
| | *---* | |
* |       | *
| *---*---* |
|           |
*---*---*---*

⭐ N Start ⭐

The default is to number points starting at N=1, exactly like the picture above. There's an optional n_start if you wanna start somewhere else, in the same pattern. Like, say, to start at 0:

n_start => 0
            45 44
         46 28 27 43
      47 29 15 14 26 42
   48 30 16  6  5 13 25 41
49 31 17  7  1  0  4 12 24 40
50 32 18  8  2  3 11 23 39 59
   51 33 19  9 10 22 38 58
      52 34 20 21 37 57
         53 35 36 56
            54 55

🧚 FUNCTIONS 🧚

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. 💫

  • $path = Math::PlanePath::AztecDiamondRings->new ()
  • $path = Math::PlanePath::AztecDiamondRings->new (n_start => $n) — creates and returns a brand new Aztec diamond spiral object. Iconic behaviour, no notes. 👑
  • ($x,$y) = $path->n_to_xy ($n) — returns the X,Y coordinates of point number $n on the path.
    • For $n < 1 the return is an empty list, because the path is considered to start at 1. Periodt. 🙈
  • $n = $path->xy_to_n ($x,$y) — returns the point number for coordinates $x,$y. Both $x and $y get rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1, so the entire plane is covered. No gaps, no crumbs. 💅
  • ($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2) — the returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle. Not approximate, not vibes-based, exact fr fr.

🔥 FORMULAS 🔥

🦋 X,Y to N

The path makes lines in each quadrant. The quadrant is determined by the signs of X and Y, then the line in that quadrant is either d=X+Y or d=X-Y. A quadratic in d gives a starting N for the line, and Y (or X if you want) is an offset from there:

Y>=0 X>=0     d=X+Y  N=(2d+2)*d+1 + Y
Y>=0 X<0      d=Y-X  N=2d^2       - Y
Y<0  X>=0     d=X-Y  N=(2d+2)*d+1 + Y
Y<0  X<0      d=X+Y  N=(2d+4)*d+2 - Y

For example:

Y=2 X=3       d=2+3=5      N=(2*5+2)*5+1  + 2  = 63
Y=2 X=-1      d=2-(-1)=3   N=2*3*3        - 2  = 16
Y=-1 X=4      d=4-(-1)=5   N=(2*5+2)*5+1  + -1 = 60
Y=-2 X=-3     d=-3+(-2)=-5 N=(2*-5+4)*-5+2 - (-2) = 34

The two X>=0 cases are the same N formula and can be combined with an abs:

X>=0          d=X+abs(Y)   N=(2d+2)*d+1 + Y

This works because at Y=0 the last line of one ring joins up to the start of the next. For example N=11 to N=15:

15             2
 \
  14          1
    \
     13   <- Y=0

  12         -1
 /
11            -2

^
X=0 1  2

💎 Rectangle to N Range

Within each row N increases as X increases away from the Y axis, and within each column N similarly increases as Y increases away from the X axis. So in a rectangle the maximum N is at one of the four corners of the rectangle.

             |
   x1,y2 M---|----M x2,y2
         |   |    |
      -------O---------
         |   |    |
         |   |    |
   x1,y1 M---|----M x1,y1
             |

For any two rows y1 and y2, the values in row y2 are all bigger than in y1 if y2>=-y1. This holds even when y1 and y2 are on the same side of the origin, ie. both positive or both negative.

For any two columns x1 and x2, the values in the part with Y>=0 are all bigger if x2>=-x1, or in the part of the columns with Y<0 it's x2>=-x1-1. So the biggest corner is at

max_y = (y2 >= -y1              ? y2 ? y1)
max_x = (x2 >= -x1 - (max_y<0)  ? x2 : x1)

The difference in the X handling for Y positive or negative is due to the quadrant ordering. When Y>=0, at X and -X the bigger N is the X negative side, but when Y<0 it's the X positive side. Sneaky little quadrant lore. 😌

A similar approach gives the minimum N in a rectangle.

min_y = / y2 if y2 < 0, and set xbase=-1
        | y1 if y1 > 0, and set xbase=0
        \ 0 otherwise,  and set xbase=0

min_x = / x2 if x2 < xbase
        | x1 if x1 > xbase
        \ xbase otherwise

The minimum row is Y=0, but if that's not in the rectangle then the y2 or y1 top or bottom edge is the minimum. Then within any row the minimum N is at xbase=0 if Y<0 or xbase=-1 if Y>=0. If that xbase is not in range then the x2 or x1 left or right edge is the minimum. 💅

👑 OEIS 👑

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A001844> (etc)

n_start=1 (the default)
  A001844    N on X axis, the centred squares 2k(k+1)+1

n_start=0
  A046092    N on X axis, 4*triangular
  A139277    N on diagonal X=Y
  A023532    abs(dY), being 0 if N=k*(k+3)/2

🌈 SEE ALSO 🌈

Math::PlanePath, Math::PlanePath::DiamondSpiral

🏠 HOME PAGE 🏠

<http://user42.tuxfamily.org/math-planepath/index.html>

📜 LICENSE 📜

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

⚠️ real talk bestie: that "without any warranty" bit is not a joke, it's the actual license, so don't come for anyone if it breaks. 💀

okay that's the whole page!! this was perl v5.32.0, dated 2021-01-23, and the page you just read is Math::...dRings(3pm) 💖 stay symmetric, flatten all four corners, and go number some rings fr fr ✨🦋💅

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